Dynamics of sliding geometrically-exact beams: large angle maneuver and parametric resonance

Abstract

We present the dynamic formulations for sliding beams that are deployed or retrieved through prismatic joints. The beams can undergo large deformation, large overall motion, with shear deformation accounted for. Until recently, the sliding beam problem has been tackled mostly under small deformation assumptions, or under quasi-static motion. Here we employ geometrically-exact beam theory. Two theoretically-equivalent formulations are proposed: A full Lagrangian version, and an Eulerian-Lagrangian version. A salient feature of the problem is that the equations of motion in both formulations are defined on time-varying spatial domain. This feature raises some complications in the computational formulation and computer implementation. We discuss in detail the transformation of the equations in the full Lagrangian formulation from a time-varying spatial domain to a constant spatial domain via the introduction of a stretched coordinate. A Galerkin projection is then applied to discretize the resulting governing partial differential equations. Even though the system does not have any rotating motion as in gyroscopic systems, the inertia operator has a weak form that can be decomposed exactly into a symmetric part and an anti-symmetric part. The distinction between the full Lagrangian formulation and the Eulerian-Lagrangian formulation from the computer implementation viewpoint is indicated. Several numerical examples — ‘spaghetti/reverse spaghetti problem,’ beam under combined sliding motion and large angle maneuver, parametric resonance — are given to illustrate the versatility of the proposed approach. The results reveal a rich dynamical behavior to be explored further in the future.